General solution of integration by parts of int(x^n*e^x)

[brandy]

Homework Statement

i have to create a general formula for integral of (x^n * e^x) dx
using whatever method i deem appropriate. (the only way i could think of is by parts)

Homework Equations

int(x^n * e^x)dx
int(uv')dx=uv-int(vu')dx

The Attempt at a Solution

i used integration by parts. so. I am having trouble with the uv part.
so far I've got
n!*e^x * (U) - int(e^x*n!)
U=? something that sums up u-u'-u''-u'''... until x is to the power of 1.

i figured out the function n-(n-1)-(n-2) etc which is = n-n(n-1)/2
i think if i can manipulate it enough it can give me the solution. but idk how

really, i just need a push in the right direction. or some clues or hints or something. ps make it simple, i take a while to understand other peoples working.

 

[brandy]

ooooh poooooo!
i just realized this doesn't factor in the fact that you end up minusing a new function with a negative in it and so on so the solution ends up as +term -term + term - term etc


also, something that just further comfused me was this:
i read the next part of the question which says that i need to derive this formula which is the solution to the problem


I_n = xⁿ e^x - n*I_n-1
where I_n \intxⁿ e^x dx

 

[rock.freak667]

I_n = xⁿ e^x - n*I_n-1
where I_n \intxⁿ e^x dx

Just apply integration by parts once to I_n and that result would easily follow.